skip to main content

PRIMA: general and precise neural network certification via scalable convex hull approximations

Published:12 January 2022Publication History
Skip Abstract Section

Abstract

Formal verification of neural networks is critical for their safe adoption in real-world applications. However, designing a precise and scalable verifier which can handle different activation functions, realistic network architectures and relevant specifications remains an open and difficult challenge.

In this paper, we take a major step forward in addressing this challenge and present a new verification framework, called PRIMA. PRIMA is both (i) general: it handles any non-linear activation function, and (ii) precise: it computes precise convex abstractions involving multiple neurons via novel convex hull approximation algorithms that leverage concepts from computational geometry. The algorithms have polynomial complexity, yield fewer constraints, and minimize precision loss.

We evaluate the effectiveness of PRIMA on a variety of challenging tasks from prior work. Our results show that PRIMA is significantly more precise than the state-of-the-art, verifying robustness to input perturbations for up to 20%, 30%, and 34% more images than existing work on ReLU-, Sigmoid-, and Tanh-based networks, respectively. Further, PRIMA enables, for the first time, the precise verification of a realistic neural network for autonomous driving within a few minutes.

Skip Supplemental Material Section

Supplemental Material

Auxiliary Presentation Video

A video of a short presentation of our work on PRIMA, a general and precise neural network certification framework leveraging scalable convex hull approximations. Formal verification of neural networks is critical for their safe adoption in real-world applications. However, designing a precise and scalable verifier remains an open and difficult challenge. In this paper, we take a major step forward in addressing this challenge and present a new verification framework, called Prima. Prima is both (i) general: it handles any non-linear activation function, and (ii) precise: it computes precise convex abstractions involving multiple neurons via novel convex hull approximation algorithms that leverage concepts from computational geometry. The algorithms have polynomial complexity, yield fewer constraints, and minimize precision loss.

References

  1. Greg Anderson, Shankara Pailoor, Isil Dillig, and Swarat Chaudhuri. 2019. Optimization and Abstraction: A Synergistic Approach for Analyzing Neural Network Robustness. In Proc. Programming Language Design and Implementation (PLDI). 731–744. https://doi.org/10.1145/3314221.3314614 Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Ross Anderson, Joey Huchette, Will Ma, Christian Tjandraatmadja, and Juan Pablo Vielma. 2020. Strong mixed-integer programming formulations for trained neural networks. Mathematical Programming, 1–37. https://doi.org/10.1007/s10107-020-01474-5 Google ScholarGoogle ScholarCross RefCross Ref
  3. David Avis and Komei Fukuda. 1991. A basis enumeration algorithm for linear systems with geometric applications. Applied Mathematics Letters, 4, 5 (1991), 39–42. https://doi.org/10.1016/0893-9659(91)90141-H Google ScholarGoogle ScholarCross RefCross Ref
  4. David Avis and Komei Fukuda. 1992. A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete & Computational Geometry, 8, 3 (1992), 295–313. https://doi.org/10.1007/BF02293050 Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Mislav Balunovic, Maximilian Baader, Gagandeep Singh, Timon Gehr, and Martin T. Vechev. 2019. Certifying Geometric Robustness of Neural Networks. In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d’Alché-Buc, Emily B. Fox, and Roman Garnett (Eds.). 15287–15297. https://proceedings.neurips.cc/paper/2019/hash/f7fa6aca028e7ff4ef62d75ed025fe76-Abstract.htmlGoogle ScholarGoogle Scholar
  6. C Bradford Barber, David P Dobkin, and Hannu Huhdanpaa. 1993. The quickhull algorithm for convex hull. Technical Report GCG53, The Geometry Center, MN. https://doi.org/10.1145/235815.235821 Google ScholarGoogle Scholar
  7. Jon Louis Bentley, Franco P Preparata, and Mark G Faust. 1982. Approximation algorithms for convex hulls. Commun. ACM, 25, 1 (1982), 64–68. https://doi.org/10.1145/358315.358392 Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Mariusz Bojarski, Davide Del Testa, Daniel Dworakowski, Bernhard Firner, Beat Flepp, Prasoon Goyal, Lawrence D Jackel, Mathew Monfort, Urs Muller, and Jiakai Zhang. 2016. End to end learning for self-driving cars. ArXiv preprint, abs/1604.07316 (2016), arxiv:1604.07316Google ScholarGoogle Scholar
  9. Akhilan Boopathy, Tsui-Wei Weng, Pin-Yu Chen, Sijia Liu, and Luca Daniel. 2019. CNN-Cert: An Efficient Framework for Certifying Robustness of Convolutional Neural Networks. In The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, January 27 - February 1, 2019. AAAI Press, 3240–3247. https://doi.org/10.1609/aaai.v33i01.33013240 Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Elena Botoeva, Panagiotis Kouvaros, Jan Kronqvist, Alessio Lomuscio, and Ruth Misener. 2020. Efficient Verification of ReLU-Based Neural Networks via Dependency Analysis. In The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New York, NY, USA, February 7-12, 2020. AAAI Press, 3291–3299. https://doi.org/10.1609/aaai.v34i04.5729 Google ScholarGoogle ScholarCross RefCross Ref
  11. Rudy Bunel, Oliver Hinder, Srinadh Bhojanapalli, and Krishnamurthy Dvijotham. 2020. An efficient nonconvex reformulation of stagewise convex optimization problems. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin (Eds.). https://proceedings.neurips.cc/paper/2020/hash/5d97f4dd7c44b2905c799db681b80ce0-Abstract.htmlGoogle ScholarGoogle Scholar
  12. Rudy Bunel, Jingyue Lu, Ilker Turkaslan, Pushmeet Kohli, P Torr, and P Mudigonda. 2020. Branch and bound for piecewise linear neural network verification. Journal of Machine Learning Research, 21, 2020 (2020).Google ScholarGoogle Scholar
  13. Bernard Chazelle. 1993. An optimal convex hull algorithm in any fixed dimension. Discrete & Computational Geometry, 10, 4 (1993), 377–409. https://doi.org/10.1007/BF02573985 Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Robert Clarisó and Jordi Cortadella. 2007. The octahedron abstract domain. Science of Computer Programming, 64, 1 (2007), 115–139. https://doi.org/10.1007/978-3-540-27864-1_23 Google ScholarGoogle ScholarCross RefCross Ref
  15. Jeremy M. Cohen, Elan Rosenfeld, and J. Zico Kolter. 2019. Certified Adversarial Robustness via Randomized Smoothing. In Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, Kamalika Chaudhuri and Ruslan Salakhutdinov (Eds.) (Proceedings of Machine Learning Research, Vol. 97). PMLR, 1310–1320. http://proceedings.mlr.press/v97/cohen19c.htmlGoogle ScholarGoogle Scholar
  16. Patrick Cousot. 1996. Abstract Interpretation. ACM Comput. Surv., 28, 2 (1996), 324–328. https://doi.org/10.1145/234528.234740 Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. George Bernard Dantzig. 1998. Linear programming and extensions. 48, Princeton university press. https://doi.org/10.1515/9781400884179 Google ScholarGoogle ScholarCross RefCross Ref
  18. Sumanth Dathathri, Krishnamurthy Dvijotham, Alexey Kurakin, Aditi Raghunathan, Jonathan Uesato, Rudy Bunel, Shreya Shankar, Jacob Steinhardt, Ian J. Goodfellow, Percy Liang, and Pushmeet Kohli. 2020. Enabling certification of verification-agnostic networks via memory-efficient semidefinite programming. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin (Eds.). https://proceedings.neurips.cc/paper/2020/hash/397d6b4c83c91021fe928a8c4220386b-Abstract.htmlGoogle ScholarGoogle Scholar
  19. Herbert Edelsbrunner. 2012. Algorithms in combinatorial geometry. 10, Springer Science & Business Media. https://doi.org/10.1007/978-3-642-61568-9 Google ScholarGoogle ScholarCross RefCross Ref
  20. Ruediger Ehlers. 2017. Formal verification of piece-wise linear feed-forward neural networks. In International Symposium on Automated Technology for Verification and Analysis. 269–286. https://doi.org/10.1007/978-3-319-68167-2_19 Google ScholarGoogle ScholarCross RefCross Ref
  21. Komei Fukuda. 2020. Polyhedral Computation. isbn:978-3-907234-10-5 https://doi.org/10.3929/ethz-b-000426218 Google ScholarGoogle ScholarCross RefCross Ref
  22. Komei Fukuda and Alain Prodon. 1995. Double description method revisited. In Franco-Japanese and Franco-Chinese Conference on Combinatorics and Computer Science. 91–111. https://doi.org/10.1007/3-540-61576-8_77 Google ScholarGoogle ScholarCross RefCross Ref
  23. Timon Gehr, Matthew Mirman, Dana Drachsler-Cohen, Petar Tsankov, Swarat Chaudhuri, and Martin Vechev. 2018. Ai2: Safety and robustness certification of neural networks with abstract interpretation. In 2018 IEEE Symposium on Security and Privacy (SP). 3–18. https://doi.org/10.1109/SP.2018.00058 Google ScholarGoogle ScholarCross RefCross Ref
  24. Blagoy Genov. 2015. The convex hull problem in practice: improving the running time of the double description method. Ph.D. Dissertation.Google ScholarGoogle Scholar
  25. Sven Gowal, Krishnamurthy Dvijotham, Robert Stanforth, Rudy Bunel, Chongli Qin, Jonathan Uesato, Relja Arandjelovic, Timothy Arthur Mann, and Pushmeet Kohli. 2019. Scalable Verified Training for Provably Robust Image Classification. In 2019 IEEE/CVF International Conference on Computer Vision, ICCV 2019, Seoul, Korea (South), October 27 - November 2, 2019. IEEE, 4841–4850. https://doi.org/10.1109/ICCV.2019.00494 Google ScholarGoogle ScholarCross RefCross Ref
  26. Gurobi Optimization, LLC. 2018. Gurobi Optimizer Reference Manual. http://www.gurobi.comGoogle ScholarGoogle Scholar
  27. Xiaowei Huang, Marta Kwiatkowska, Sen Wang, and Min Wu. 2017. Safety verification of deep neural networks. In International Conference on Computer Aided Verification. 3–29. https://doi.org/10.1007/978-3-319-63387-9_1 Google ScholarGoogle Scholar
  28. Michael Joswig. 2003. Beneath-and-beyond revisited. In Algebra, Geometry and Software Systems. Springer, 1–21. https://doi.org/10.1007/978-3-662-05148-1_1 Google ScholarGoogle ScholarCross RefCross Ref
  29. Guy Katz, Clark Barrett, David L Dill, Kyle Julian, and Mykel J Kochenderfer. 2017. Reluplex: An efficient SMT solver for verifying deep neural networks. In International Conference on Computer Aided Verification. 97–117. https://doi.org/10.1007/978-3-319-63387-9_5 Google ScholarGoogle Scholar
  30. Guy Katz, Derek A Huang, Duligur Ibeling, Kyle Julian, Christopher Lazarus, Rachel Lim, Parth Shah, Shantanu Thakoor, Haoze Wu, and Aleksandar Zeljić. 2019. The marabou framework for verification and analysis of deep neural networks. In International Conference on Computer Aided Verification. 443–452. https://doi.org/10.1007/978-3-030-25540-4_26 Google ScholarGoogle ScholarCross RefCross Ref
  31. Hamid R Khosravani, António E Ruano, and Pedro M Ferreira. 2013. A simple algorithm for convex hull determination in high dimensions. In 2013 IEEE 8th International Symposium on Intelligent Signal Processing. 109–114. https://doi.org/10.1109/WISP.2013.6657492 Google ScholarGoogle ScholarCross RefCross Ref
  32. Mathias Lecuyer, Vaggelis Atlidakis, Roxana Geambasu, Daniel Hsu, and Suman Jana. 2018. Certified Robustness to Adversarial Examples with Differential Privacy. 2019 IEEE Symposium on Security and Privacy (S&P), https://doi.org/10.1109/SP.2019.00044 Google ScholarGoogle ScholarCross RefCross Ref
  33. Jingyue Lu and M. Pawan Kumar. 2020. Neural Network Branching for Neural Network Verification. In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020. OpenReview.net. https://openreview.net/forum?id=B1evfa4tPBGoogle ScholarGoogle Scholar
  34. Zhaoyang Lyu, Ching-Yun Ko, Zhifeng Kong, Ngai Wong, Dahua Lin, and Luca Daniel. 2020. Fastened CROWN: Tightened Neural Network Robustness Certificates. In The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New York, NY, USA, February 7-12, 2020. AAAI Press, 5037–5044. https://doi.org/10.1609/aaai.v34i04.5944 Google ScholarGoogle ScholarCross RefCross Ref
  35. Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. 2018. Towards Deep Learning Models Resistant to Adversarial Attacks. In 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Conference Track Proceedings. OpenReview.net. https://openreview.net/forum?id=rJzIBfZAbGoogle ScholarGoogle Scholar
  36. Alexandre Maréchal and Michaël Périn. 2017. Efficient elimination of redundancies in polyhedra using raytracing.Google ScholarGoogle Scholar
  37. Matthew Mirman, Timon Gehr, and Martin T. Vechev. 2018. Differentiable Abstract Interpretation for Provably Robust Neural Networks. In Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholmsmässan, Stockholm, Sweden, July 10-15, 2018, Jennifer G. Dy and Andreas Krause (Eds.) (Proceedings of Machine Learning Research, Vol. 80). PMLR, 3575–3583. http://proceedings.mlr.press/v80/mirman18b.htmlGoogle ScholarGoogle Scholar
  38. David R Morrison, Sheldon H Jacobson, Jason J Sauppe, and Edward C Sewell. 2016. Branch-and-bound algorithms: A survey of recent advances in searching, branching, and pruning. Discrete Optimization, 19 (2016), 79–102. https://doi.org/10.1016/j.disopt.2016.01.005 Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. Theodore S Motzkin, Howard Raiffa, Gerald L Thompson, and Robert M Thrall. 1953. The double description method. Contributions to the Theory of Games, 2, 28 (1953), 51–73. https://doi.org/10.1515/9781400881970-004 Google ScholarGoogle ScholarCross RefCross Ref
  40. Christoph Müller, Francois Serre, Gagandeep Singh, Markus Püschel, and Martin Vechev. 2021. Scaling Polyhedral Neural Network Verification on GPUs. Proc. Machine Learning and Systems (MLSys).Google ScholarGoogle Scholar
  41. Christoph Müller, Gagandeep Singh, Markus Püschel, and Martin Vechev. 2020. Neural Network Robustness Verification on GPUs. arxiv:cs.LG/2007.10868.Google ScholarGoogle Scholar
  42. Alessandro De Palma, Harkirat S. Behl, Rudy R. Bunel, Philip H. S. Torr, and M. Pawan Kumar. 2021. Scaling the Convex Barrier with Active Sets. In 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. OpenReview.net. https://openreview.net/forum?id=uQfOy7LrlTRGoogle ScholarGoogle Scholar
  43. Aditi Raghunathan, Jacob Steinhardt, and Percy Liang. 2018. Semidefinite relaxations for certifying robustness to adversarial examples. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Garnett (Eds.). 10900–10910. https://proceedings.neurips.cc/paper/2018/hash/29c0605a3bab4229e46723f89cf59d83-Abstract.htmlGoogle ScholarGoogle Scholar
  44. Anian Ruoss, Maximilian Baader, Mislav Balunović, and Martin Vechev. 2020. Efficient Certification of Spatial Robustness. ArXiv preprint, abs/2009.09318 (2020), arxiv:2009.09318Google ScholarGoogle Scholar
  45. Anian Ruoss, Mislav Balunovic, Marc Fischer, and Martin T. Vechev. 2020. Learning Certified Individually Fair Representations. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin (Eds.). https://proceedings.neurips.cc/paper/2020/hash/55d491cf951b1b920900684d71419282-Abstract.htmlGoogle ScholarGoogle Scholar
  46. Wonryong Ryou, Jiayu Chen, Mislav Balunovic, Gagandeep Singh, Andrei Dan, and Martin Vechev. 2020. Fast and effective robustness certification for recurrent neural networks. ArXiv preprint, abs/2005.13300 (2020), arxiv:2005.13300Google ScholarGoogle Scholar
  47. Hadi Salman, Jerry Li, Ilya P. Razenshteyn, Pengchuan Zhang, Huan Zhang, Sébastien Bubeck, and Greg Yang. 2019. Provably Robust Deep Learning via Adversarially Trained Smoothed Classifiers. In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d’Alché-Buc, Emily B. Fox, and Roman Garnett (Eds.). 11289–11300. https://proceedings.neurips.cc/paper/2019/hash/3a24b25a7b092a252166a1641ae953e7-Abstract.htmlGoogle ScholarGoogle Scholar
  48. Hadi Salman, Greg Yang, Huan Zhang, Cho-Jui Hsieh, and Pengchuan Zhang. 2019. A Convex Relaxation Barrier to Tight Robustness Verification of Neural Networks. In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d’Alché-Buc, Emily B. Fox, and Roman Garnett (Eds.). 9832–9842. https://proceedings.neurips.cc/paper/2019/hash/246a3c5544feb054f3ea718f61adfa16-Abstract.htmlGoogle ScholarGoogle Scholar
  49. Hossein Sartipizadeh and Tyrone L Vincent. 2016. Computing the approximate convex hull in high dimensions. ArXiv preprint, abs/1603.04422 (2016), arxiv:1603.04422Google ScholarGoogle Scholar
  50. Raimund Seidel. 1995. The upper bound theorem for polytopes: an easy proof of its asymptotic version. Computational Geometry, 5, 2 (1995), 115–116. https://doi.org/10.1016/0925-7721(95)00013-Y Google ScholarGoogle ScholarDigital LibraryDigital Library
  51. Gagandeep Singh, Rupanshu Ganvir, Markus Püschel, and Martin T. Vechev. 2019. Beyond the Single Neuron Convex Barrier for Neural Network Certification. In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d’Alché-Buc, Emily B. Fox, and Roman Garnett (Eds.). 15072–15083. https://proceedings.neurips.cc/paper/2019/hash/0a9fdbb17feb6ccb7ec405cfb85222c4-Abstract.htmlGoogle ScholarGoogle Scholar
  52. Gagandeep Singh, Timon Gehr, Matthew Mirman, Markus Püschel, and Martin T. Vechev. 2018. Fast and Effective Robustness Certification. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Garnett (Eds.). 10825–10836. https://proceedings.neurips.cc/paper/2018/hash/f2f446980d8e971ef3da97af089481c3-Abstract.htmlGoogle ScholarGoogle Scholar
  53. Gagandeep Singh, Timon Gehr, Markus Püschel, and Martin Vechev. 2019. An abstract domain for certifying neural networks. Proceedings of the ACM on Programming Languages, 3, POPL (2019), 1–30. https://doi.org/10.1145/3290354 Google ScholarGoogle ScholarDigital LibraryDigital Library
  54. Gagandeep Singh, Timon Gehr, Markus Püschel, and Martin T. Vechev. 2019. Boosting Robustness Certification of Neural Networks. In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net. https://openreview.net/forum?id=HJgeEh09KQGoogle ScholarGoogle Scholar
  55. Gagandeep Singh, Markus Püschel, and Martin Vechev. 2017. Fast Polyhedra Abstract Domain. In Proc. Principles of Programming Languages (POPL). 46–59. https://doi.org/10.1145/3009837.3009885 Google ScholarGoogle ScholarDigital LibraryDigital Library
  56. Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian J. Goodfellow, and Rob Fergus. 2014. Intriguing properties of neural networks. In 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings, Yoshua Bengio and Yann LeCun (Eds.). arxiv:1312.6199Google ScholarGoogle Scholar
  57. Christian Tjandraatmadja, Ross Anderson, Joey Huchette, Will Ma, Krunal Patel, and Juan Pablo Vielma. 2020. The Convex Relaxation Barrier, Revisited: Tightened Single-Neuron Relaxations for Neural Network Verification. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin (Eds.). https://proceedings.neurips.cc/paper/2020/hash/f6c2a0c4b566bc99d596e58638e342b0-Abstract.htmlGoogle ScholarGoogle Scholar
  58. Vincent Tjeng, Kai Y. Xiao, and Russ Tedrake. 2019. Evaluating Robustness of Neural Networks with Mixed Integer Programming. In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net. https://openreview.net/forum?id=HyGIdiRqtmGoogle ScholarGoogle Scholar
  59. Udacity. 2016. Using Deep Learning to Predict Steering Angles. https://github.com/udacity/self-driving-carGoogle ScholarGoogle Scholar
  60. Caterina Urban and Antoine Miné. 2021. A Review of Formal Methods applied to Machine Learning. ArXiv preprint, abs/2104.02466 (2021), arxiv:2104.02466Google ScholarGoogle Scholar
  61. Shiqi Wang, Kexin Pei, Justin Whitehouse, Junfeng Yang, and Suman Jana. 2018. Efficient Formal Safety Analysis of Neural Networks. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Garnett (Eds.). 6369–6379. https://proceedings.neurips.cc/paper/2018/hash/2ecd2bd94734e5dd392d8678bc64cdab-Abstract.htmlGoogle ScholarGoogle Scholar
  62. Shiqi Wang, Huan Zhang, Kaidi Xu, Xue Lin, Suman Jana, Cho-Jui Hsieh, and J Zico Kolter. 2021. Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Verification. ArXiv preprint, abs/2103.06624 (2021), arxiv:2103.06624Google ScholarGoogle Scholar
  63. Tsui-Wei Weng, Huan Zhang, Hongge Chen, Zhao Song, Cho-Jui Hsieh, Luca Daniel, Duane S. Boning, and Inderjit S. Dhillon. 2018. Towards Fast Computation of Certified Robustness for ReLU Networks. In Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholmsmässan, Stockholm, Sweden, July 10-15, 2018, Jennifer G. Dy and Andreas Krause (Eds.) (Proceedings of Machine Learning Research, Vol. 80). PMLR, 5273–5282. http://proceedings.mlr.press/v80/weng18a.htmlGoogle ScholarGoogle Scholar
  64. Eric Wong, Frank R. Schmidt, Jan Hendrik Metzen, and J. Zico Kolter. 2018. Scaling provable adversarial defenses. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Garnett (Eds.). 8410–8419. https://proceedings.neurips.cc/paper/2018/hash/358f9e7be09177c17d0d17ff73584307-Abstract.htmlGoogle ScholarGoogle Scholar
  65. Weiming Xiang, Hoang-Dung Tran, and Taylor T Johnson. 2018. Output reachable set estimation and verification for multilayer neural networks. IEEE transactions on neural networks and learning systems, 29, 11 (2018), 5777–5783. https://doi.org/10.1109/TNNLS.2018.2808470 Google ScholarGoogle ScholarCross RefCross Ref
  66. Kaidi Xu, Zhouxing Shi, Huan Zhang, Yihan Wang, Kai-Wei Chang, Minlie Huang, Bhavya Kailkhura, Xue Lin, and Cho-Jui Hsieh. 2020. Automatic Perturbation Analysis for Scalable Certified Robustness and Beyond. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin (Eds.). https://proceedings.neurips.cc/paper/2020/hash/0cbc5671ae26f67871cb914d81ef8fc1-Abstract.htmlGoogle ScholarGoogle Scholar
  67. Kaidi Xu, Huan Zhang, Shiqi Wang, Yihan Wang, Suman Jana, Xue Lin, and Cho-Jui Hsieh. 2021. Fast and Complete: Enabling Complete Neural Network Verification with Rapid and Massively Parallel Incomplete Verifiers. In 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. OpenReview.net. https://openreview.net/forum?id=nVZtXBI6LNnGoogle ScholarGoogle Scholar
  68. Zong-Ben Xu, Jiang-She Zhang, and Yiu-Wing Leung. 1998. An approximate algorithm for computing multidimensional convex hulls. Applied mathematics and computation, 94, 2-3 (1998), 193–226. https://doi.org/10.1016/S0096-3003(97)10043-1 Google ScholarGoogle ScholarDigital LibraryDigital Library
  69. Huan Zhang, Tsui-Wei Weng, Pin-Yu Chen, Cho-Jui Hsieh, and Luca Daniel. 2018. Efficient Neural Network Robustness Certification with General Activation Functions. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Garnett (Eds.). 4944–4953. https://proceedings.neurips.cc/paper/2018/hash/d04863f100d59b3eb688a11f95b0ae60-Abstract.htmlGoogle ScholarGoogle Scholar
  70. Jinhong Zhong, Ke Tang, and A Kai Qin. 2014. Finding convex hull vertices in metric space. In 2014 International Joint Conference on Neural Networks (IJCNN). 1587–1592. https://doi.org/10.1109/IJCNN.2014.6889699 Google ScholarGoogle ScholarCross RefCross Ref

Index Terms

  1. PRIMA: general and precise neural network certification via scalable convex hull approximations

        Recommendations

        Comments

        Login options

        Check if you have access through your login credentials or your institution to get full access on this article.

        Sign in

        Full Access

        PDF Format

        View or Download as a PDF file.

        PDF

        eReader

        View online with eReader.

        eReader
        About Cookies On This Site

        We use cookies to ensure that we give you the best experience on our website.

        Learn more

        Got it!